# Bishop's compactness theorem and convergence of analytic subset

Let $$V_i$$ be a sequence of $$k$$ dimensional analytic subsets in $$\mathbb C^N$$. Suppose that the volume of $$V_i$$ is uniformly bounded, then Bishop's compactness theorem says that $$V_i$$ will convergence by sequence to an analytic subsets $$V$$.

Q1: What is the precise meaning of "converge" here. Q2: Is it possible that a sequence of singular point $$q_i\in V_i$$ converge to a smooth point $$q\in V$$. It seems impossible.

For the second, it is easy to construct an example of singular spaces converging to smooth. Take a sequence of curves in $${\Bbb C}P^2$$ with each curve obtained as a union of two projective lines. Assume that this sequence converges to a union of a line with itself. Then the limit is smooth.